As for other browsers: I guess that you'd still be able to access my website, but YouTube and GMail will just show you a generic "This browser is not supported" or, at best, "Your browser had been deemed insecure. Please use a secure browser".
If we go full Nostradamus the most likely result is that Safari will play ball (because they are already doing something similar), Firefox will play ball (because they no longer have a spine), they'll implement whatever Google says, and yet another portion of the web will be closed to those who don't dance to Google's tune. I hope you're not too attached to youtube-dl or your adblocker.
The problem is given the number of Google's services, it can easily "nudge" site owners into implementing it.
They can make it mandatory for adwords or analytics or any other service that they have for webmasters and everyone will have to jump in while it is still technically voluntary.
If your $bank choses to implement it, then it can very well mandate only windows or osx clients. Then you can't use firefox on linux.
This may sound like conspiracy theory but we already have examples of similar restrictions on android apps with safetynet. Bank apps already refuse to run on custom roms. It will not be limited to banks. Someone mentioned that McDonald's app doesn't work unless you pass safetynet.
That is how ridiculous it can become.
Edit: fixed typos
"Sorry, this site requires a certified web browser.
Your browser does not meet the security requirements that this website requires.
Please upgrade to the lastest Google Chrome and remove any uncertified plugins."